Theorem imaeq2 5002

Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Assertion

Ref	Expression
imaeq2	|- ( A = B -> ( C " A ) = ( C " B ) )

Proof of Theorem imaeq2

Step	Hyp	Ref	Expression
1		reseq2 4938	. . 3 |- ( A = B -> ( C |` A ) = ( C |` B ) )
2	1	rneqd 4892	. 2 |- ( A = B -> ran ( C |` A ) = ran ( C |` B ) )
3		df-ima 4673	. 2 |- ( C " A ) = ran ( C |` A )
4		df-ima 4673	. 2 |- ( C " B ) = ran ( C |` B )
5	2, 3, 4	3eqtr4g 2251	1 |- ( A = B -> ( C " A ) = ( C " B ) )

Syntax hints:   -> wi 4   = wceq 1364   ran crn 4661   |` cres 4662   "cima 4663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-cnv 4668  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673

This theorem is referenced by:  imaeq2i  5004  imaeq2d  5006  fimadmfo  5486  ssimaex  5619  ssimaexg  5620  isoselem  5864  f1opw2  6126  fopwdom  6894  ssenen  6909  fiintim  6987  fidcenumlemrk  7015  fidcenumlemr  7016  sbthlem2  7019  isbth  7028  ennnfonelemp1  12566  ennnfonelemnn0  12582  ctinfomlemom  12587  ctinfom  12588  tgcn  14387  iscnp4  14397  cnpnei  14398  cnima  14399  cnconst2  14412  cnrest2  14415  cnptoprest  14418  txcnp  14450  txcnmpt  14452  metcnp3  14690